A284894 Positions of 0 in A284893; complement of A284895.

1, 3, 7, 9, 13, 17, 21, 23, 27, 29, 33, 37, 41, 43, 47, 51, 55, 57, 61, 65, 69, 71, 75, 77, 81, 85, 89, 91, 95, 97, 101, 105, 109, 111, 115, 119, 123, 125, 129, 133, 137, 139, 143, 145, 149, 153, 157, 159, 163, 167, 171, 173, 177, 181, 185, 187, 191, 193

Offset

1,2

Comments

Conjecture: 2 < n*r - a(n) < 6 for n >= 1, where r = 2 + sqrt(2).

Let (d(n)) be the sequence of first differences of (a(n)), i.e., d(n)=a(n+1)-a(n) for n=1,2,... Let sigma be the morphism fixing (x(n)), where x = A284893. Then x is a concatenation of words B(i), where B(i) is either sigma(0)=01 or sigma(1)=0111. The crucial observation is that d(n+1)=2 if and only if B(n)=01, and d(n+1) =4 if and only if B(n)=0111. It follows that (d(n)) = 2,4,2,4,4,4,... is the unique fixed point of the morphism tau: 2->24, 4->2444 (tau equals sigma up to a change of alphabet). - Michel Dekking, Jan 16 2018

Since the positions of 0 in x are partial sums of the terms of d, one can prove, using the Perron Frobenius theorem, that a weak form of the conjecture above holds: the sequence (n*r - a(n)) is bounded. This result can also be derived from the simple part (the second eigenvalue 2-sqrt(2) of the incidence matrix of the morphism tau: 2->24, 4->2444 is smaller than 1) of Theorem 1 in the paper "Symbolic discrepancy and self-similar dynamics" by Boris Adamczewski. - Michel Dekking, Jan 16 2018

Links

Clark Kimberling, Table of n, a(n) for n = 1..10000

B. Adamczewski, Symbolic discrepancy and self-similar dynamics, Annales de l'Institut Fourier 54 (2004), 2201-2234.

Example

As a word, A284893 = 010111010..., in which 0 is in positions 1,3,7,9,...

Mathematica

s = Nest[Flatten[# /. {0 -> {0, 1}, 1 -> {0, 1, 1, 1}}] &, {0}, 6] (* A284893 *)
Flatten[Position[s, 0]]  (* A284894 *)
Flatten[Position[s, 1]]  (* A284895 *)

Crossrefs

Cf. A284893, A284895.

Sequence in context: A284884 A130568 A143803 * A020497 A023490 A032375

Adjacent sequences: A284891 A284892 A284893 * A284895 A284896 A284897

Keyword

nonn,easy

Author

Clark Kimberling, Apr 16 2017

Status

approved